# Index is multiplicative for profinite groups

Suppose $G$ is a profinite group and $K,H$ are closed subgroups of $G$ with $K \le H$. Note that $K$ automatically becomes closed in $H$. (Conversely, $K$ being closed in $H$ and $H$ being closed in $G$ would imply that $K$ is closed in $G$). Then, we have:
$[G:K] = [G:H][H:K]$
where $[G:K]$ denote the respective values for the index of a closed subgroup in a profinite group where the subgroup is $K$ and the group is $G$. Similarly for $[G:H]$ (index of subgroup $H$ in group $G$) and $[H:K]$ (index of subgroup $K$ in group $H$).